Quantum symmetries of face models and the double triangle algebra

Trinchero, Roberto

doi:10.4310/atmp.2006.v10.n1.a3

Cited by 6 publications

(7 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…15. We shall not discuss here the structure of B, the interested reader may look at [30,1,36,5,43], but notice that these 6j symbols have four labels for the simple objects of type E (the module), one label for the type A and one label for the type O (the quantum symmetries), whereas the 6j symbols appearing in Fig. 14 have three indices of type E and three indices of type A.…”

Section: Coupling Of Su (2) To Ade Mattermentioning

confidence: 99%

Notes on TQFT Wire Models and Coherence Equations for SU(3) Triangular Cells

Coquereaux¹

2010

SIGMA

View full text Add to dashboard Cite

Abstract. After a summary of the TQFT wire model formalism we bridge the gap from Kuperberg equations for SU (3) spiders to Ocneanu coherence equations for systems of triangular cells on fusion graphs that describe modules associated with the fusion category of SU (3) at level k. We show how to solve these equations in a number of examples.Key words: quantum symmetries; module-categories; conformal field theories; 6j symbols 2010 Mathematics Subject Classification: 81R50; 81R10; 20C08; 18D10 ForewordStarting with the collection of irreducible integrable representations (irreps) of SU (3) at some level k (constructed in the framework of affine algebras or in the framework of quantum groups at roots of unity), the problem is to decide whether a graph encoding the action of these irreps actually defines a "healthy fusion graph" associated with a bona fide SU (3) nimrep, i.e. a module over a particular kind of fusion category. This is done by associating a complex number (a "triangular cell") with every elementary triangle of the given graph in such a way that their collection (a "self-connection") obeys a system of non trivial quadratic and quartic equations, called Ocneanu coherence equations, that can be themselves derived from another set of equations (sometimes called Kuperberg equations) describing relations between the intertwiners of the underlying fusion category. One issue is to describe and derive the coherence equations themselves. Another issue is to use them on the family of examples giving rise to SU (3) nimreps. Both problems are studied in this article.Our paper consists of two largely independent parts. The first is a set of notes dealing with TQFT graphical models (wire models, spiders, etc.). Our motivation for this part was to show how the Ocneanu coherence equations for triangular cells of fusions graphs could be deduced from the so-called Kuperberg relations for SU (3), something that does not seem to be explained in the literature. This section was then enlarged in order to set the discussion in the larger framework of graphical TQFT models and to discuss some not so well known features that show up when comparing the SU (2) and SU (3) situations. Despite its size, this first part should not be considered as a general presentation of the subject starting from first principles, this would require a book, not an article. The second part of the paper deals about the coherence equations themselves. In particular we show, on a selection of examples chosen among quantum subgroups (module-categories) of type SU (3), how to solve them in order to obtain a self-connection on the corresponding fusion graphs.

show abstract

Section: Coupling Of Su (2) To Ade Mattermentioning

confidence: 99%

Notes on TQFT Wire Models and Coherence Equations for SU(3) Triangular Cells

Coquereaux¹

2010

SIGMA

View full text Add to dashboard Cite

show abstract

“…There exist several interpretations of fusion coefficients (more generally of coefficients of nimreps) in terms of combinatorial constructions associated with fusion graphs: essential paths [24] (or generalizations of the latter), admissible triangles [25], (generalized) preprojective algebras or quivers [26], and they can also be used to define interesting weak Hopf algebras [27][28][29]. The translation of the sum rules involving the fusion coefficients (or those of the nimreps) into these different languages and points of view is left as an exercise to the reader.…”

Section: On the Path Matrix X And Its Spectral Propertiesmentioning

confidence: 99%

On sums of tensor and fusion multiplicities

Coquereaux¹,

Zuber²

2011

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

The total multiplicity in the decomposition into irreducibles of the tensor product λ ⊗ µ of two irreducible representations of a simple Lie algebra is invariant under conjugation of one of them ν N ν λµ = ν N ν λµ . This also applies to the fusion multiplicities of affine algebras in conformal WZW theories. In that context, the statement is equivalent to a property of the modular S matrix, viz Σ(κ) := λ S λκ = 0 if κ is a complex representation. Curiously, this vanishing of Σ(κ) also holds when κ is a quaternionic representation. We provide proofs of all these statements. These proofs rely on a case-by-case analysis, maybe overlooking some hidden symmetry principle. We also give various illustrations of these properties in the contexts of boundary conformal field theories, integrable quantum field theories and topological field theories.

show abstract

“…"Double triangle algebras" were also discussed and recognized as quantum groupoids in reference [14], where many proofs were given, starting from postulated properties of cells, taken from the axiomatics discussed in reference [32]. We can also mention [40] that analyses these algebras in terms of symmetries of faces models, and one section of [39], where an explicit calculation is carried out, relying on cell values obtained by explicit composition of basic cells.…”

Section: Discussionmentioning

confidence: 99%